The ionization and generalized ionization conjectures
The ionization and generalized ionization conjectures are the subject of two manuscripts dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math). The ionization conjecture for the nonrelativistic Coulomb model asks that the number of electrons a system of nuclei can bind exceed the total nuclear charge by at most a bounded amount, and that the ionization energy and the radius of a neutral atom stay bounded independently of the nuclear charge. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
The original result
- Excess charge and atomic bounds. A molecule with M fixed nuclei of real charges at least one and total nuclear charge Z strictly binds at most Z + CM electrons, with C universal, for the full nonrelativistic Coulomb Hamiltonian with two spin states. For neutral atoms with integer nuclear charge Z ≥ 1, the first ionization energy and, for every ground state, the radius outside which one half of an electron remains are bounded above and below by positive universal constants. Source: the manuscript Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms, dated 24 September 2026.
- Generalized ionization energies. The energy needed to remove m electrons from a neutral atom is asymptotic to aTFm7/3 whenever m → ∞ and Z/m → ∞, with the Thomas–Fermi constant; the manuscript also states both iterated limits. Source: the manuscript Generalized ionization energies for full Coulomb atoms, dated 24 September 2026.
- Formalization, as described by the repository. The scope note lean/docs/263.md states that the Lean formalization gives the large-ionization asymptotics of the generalized ionization energies and the asymptotic law for generalized outer-electron radii of neutral atoms, with the large-nuclear-charge limit taken first. It states that ground states are inputs to the radius statement and that their existence, fixed-m convergence and ground-state attainment are outside the formal statements. The scope note does not list the excess-charge bound or the uniform atomic bounds among the formal results.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 15 October 2026) |
|---|---|---|---|---|
2026-10-14 (latest commit bc31d29) |
Yao Xu, cadamcat/atomic-ionization-lean4 (GitHub) | Formalization (Lean) | A Lean 4 proof, following the first manuscript's Theorem 1.2 and Corollary 1.3 and the second manuscript's Theorem 1.1, that the first ionization energy of every neutral atom lies between two positive universal constants, that every neutral atom has a normalized ground state, that the half-electron radius of every normalized neutral ground state lies between two positive universal constants, and that the generalized ionization energies have the Thomas–Fermi asymptotics. The README states that OpenAI's Coulomb Lean libraries are included unchanged and that the model is copied from OpenAI's formal statements. The README's list of results does not include the molecular excess-charge bound. | Author-reported; the README states that the work was developed with AI assistance. The repository's verification notes record a fresh-clone build, a kernel replay and a Comparator check with a second kernel for version 1.0.0. |
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.